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A minimally informative likelihood approach to Bayesian inference and decision analysis Yuan, Ao

Abstract

For a given prior density, we minimize the Shannon Mutual Information between a parameter and the data, over a class of likelihoods defined by bounding a Bayes risk by a 'distortion parameter'. This gives a conditional distribution for the data given the parameter which provides optimal data compression, or equivalently, is minimally informative for a type of location parameter. These optimal likelihoods cannot, in general, be obtained in closed form. However, they can be found numerically. Moreover, we give two statistical senses in which the optimal likelihoods form parametric families which make the weakest possible assumptions on the data generating mechanism. In addition, we establish properties of this parametric family that characterize its behavior as the distortion parameter varies. We argue that the parametric families identified here may lead to a default technique for some settings in initial data analysis. We partially characterize the settings in which our techniques may be expected to provide useful answers. In particular, we argue that if one is interested in performing certain Bayesian hypothesis tests on a parameter that locates a typical region for the response, then our technique may provide weak but nevertheless useful inferences. We also investigated the robustness of inferences to modeling strategies for paired, blocked data.

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